Abstract

An approximate analytic method is presented that permits the calculation of all the output waves, both backward and forward traveling, for slanted, transmission-type volume gratings replayed at the Bragg angle and perfectly index matched to their surroundings. The predictions of the resulting analytic expressions are compared with numerical results produced by using the rigorous coupled-wave method and found to be accurate over a large range of parameters to the first order of the permittivity variation of the volume grating, all second-order terms being assumed negligible.

© 1992 Optical Society of America

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References

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  1. R. R. A. Syms, Practical Volume Holography (Clarendon, Oxford, 1990).
  2. A. Ramsbottom, “Characteristics and properties of powered reflection holographic optical elements,” Opto-electronics 5, 606–614 (1990).
  3. J. T. Sheridan, L. Solymar, “Boundary diffraction coefficients for calculating spurious beams produced by volume gratings,”IEE Electr. Lett. 26, 1840–1841 (1990).
    [CrossRef]
  4. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. Inst. Electr. Eng. 73, 894–937 (1985).
    [CrossRef]
  5. G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves, IEE Electromagnetic Wave Series (Peregrinus, London, 1976).
  6. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2945 (1969).
  7. R. R. A. Syms, L. Solymar, “Planar volume phase holograms formed in bleached photographic emulsions,” Appl. Opt. 22, 1479–1496 (1983).
    [CrossRef] [PubMed]
  8. M. V. Vasnetsov, M. S. Soskin, V. B. Taranenko, “Grazing diffraction by volume phase gratings,” Opt. Acta 32, 891–899 (1985).
    [CrossRef]
  9. K. A. Kong, “Second order coupled-mode equations for spatially periodic media,”J. Opt. Soc. Am. 67, 825–829 (1977).
    [CrossRef]

1990 (2)

A. Ramsbottom, “Characteristics and properties of powered reflection holographic optical elements,” Opto-electronics 5, 606–614 (1990).

J. T. Sheridan, L. Solymar, “Boundary diffraction coefficients for calculating spurious beams produced by volume gratings,”IEE Electr. Lett. 26, 1840–1841 (1990).
[CrossRef]

1985 (2)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. Inst. Electr. Eng. 73, 894–937 (1985).
[CrossRef]

M. V. Vasnetsov, M. S. Soskin, V. B. Taranenko, “Grazing diffraction by volume phase gratings,” Opt. Acta 32, 891–899 (1985).
[CrossRef]

1983 (1)

1977 (1)

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2945 (1969).

Gaylord, T. K.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. Inst. Electr. Eng. 73, 894–937 (1985).
[CrossRef]

James, G. L.

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves, IEE Electromagnetic Wave Series (Peregrinus, London, 1976).

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2945 (1969).

Kong, K. A.

Moharam, M. G.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. Inst. Electr. Eng. 73, 894–937 (1985).
[CrossRef]

Ramsbottom, A.

A. Ramsbottom, “Characteristics and properties of powered reflection holographic optical elements,” Opto-electronics 5, 606–614 (1990).

Sheridan, J. T.

J. T. Sheridan, L. Solymar, “Boundary diffraction coefficients for calculating spurious beams produced by volume gratings,”IEE Electr. Lett. 26, 1840–1841 (1990).
[CrossRef]

Solymar, L.

J. T. Sheridan, L. Solymar, “Boundary diffraction coefficients for calculating spurious beams produced by volume gratings,”IEE Electr. Lett. 26, 1840–1841 (1990).
[CrossRef]

R. R. A. Syms, L. Solymar, “Planar volume phase holograms formed in bleached photographic emulsions,” Appl. Opt. 22, 1479–1496 (1983).
[CrossRef] [PubMed]

Soskin, M. S.

M. V. Vasnetsov, M. S. Soskin, V. B. Taranenko, “Grazing diffraction by volume phase gratings,” Opt. Acta 32, 891–899 (1985).
[CrossRef]

Syms, R. R. A.

Taranenko, V. B.

M. V. Vasnetsov, M. S. Soskin, V. B. Taranenko, “Grazing diffraction by volume phase gratings,” Opt. Acta 32, 891–899 (1985).
[CrossRef]

Vasnetsov, M. V.

M. V. Vasnetsov, M. S. Soskin, V. B. Taranenko, “Grazing diffraction by volume phase gratings,” Opt. Acta 32, 891–899 (1985).
[CrossRef]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2945 (1969).

IEE Electr. Lett. (1)

J. T. Sheridan, L. Solymar, “Boundary diffraction coefficients for calculating spurious beams produced by volume gratings,”IEE Electr. Lett. 26, 1840–1841 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (1)

M. V. Vasnetsov, M. S. Soskin, V. B. Taranenko, “Grazing diffraction by volume phase gratings,” Opt. Acta 32, 891–899 (1985).
[CrossRef]

Opto-electronics (1)

A. Ramsbottom, “Characteristics and properties of powered reflection holographic optical elements,” Opto-electronics 5, 606–614 (1990).

Proc. Inst. Electr. Eng. (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. Inst. Electr. Eng. 73, 894–937 (1985).
[CrossRef]

Other (2)

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves, IEE Electromagnetic Wave Series (Peregrinus, London, 1976).

R. R. A. Syms, Practical Volume Holography (Clarendon, Oxford, 1990).

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Figures (9)

Fig. 1
Fig. 1

Schematic representation of the input and the output waves.

Fig. 2
Fig. 2

Ewald diagram showing the wave vectors of the two forward-traveling beams.

Fig. 3
Fig. 3

(a) Waves that can be generated by the z = 0 boundary with the A0,in wave incident. (b) Reflections from the I–II and the II–III boundaries.

Fig. 4
Fig. 4

Waves traveling inside the volume grating. The R0I backward-traveling wave has only one component produced by the z = d boundary. The R1I beam has two contributions from the opposite boundaries.

Fig. 5
Fig. 5

Path difference of wave 1 accumulated by traversing twice the volume grating.

Fig. 6
Fig. 6

Intensities of the two backward-traveling spurious waves (a) |R0I/A0,in|2 and (b) |R1I/A0,in|2 plotted against d/λ, the grating thickness in wavelengths. The grating recording waves are incident at θ0 = −20° and θ1 = +80°. The grating is replayed on Bragg at θ0 = −20°. ɛm = 0.02. Solid curves: rigorous theory; dashed curves: approximate theory.

Fig. 7
Fig. 7

Intensities of the two backward-traveling spurious waves (a) |R0I/A0,in|2 and (b) |R1I/A0,in|2 plotted against d/λ. Recording angles θ0 = −20° and θ1 = +80°; replay angle θ0 = −20°. ɛm = 0.05. Solid curves: rigorous theory; dashed curves: approximate theory.

Fig. 8
Fig. 8

(a) |R0I/A0,in|2 and (b) |R1I/A0,in|2 plotted against d/λ. Recording angles θ0 = −20° and θ1 = +80°; replay angle θ0 = 20. ɛm = 0.1. Solid curves: rigorous theory; dashed curves: approximate theory.

Fig. 9
Fig. 9

(a) |R0I/A0,in|2 and (b) |R1I/A0,in|2 plotted against d/λ. Recording angles θ0 = −20° and θ1 = +50°; replay angle θ0 = −20°. ɛm = 0.1. Solid curves: rigorous theory; dashed curves: approximate theory.

Equations (35)

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ɛ I I = 1 + ɛ m cos ( K · r ) = 1 + ɛ m cos ( K x x + K z z ) ,
ρ 0 = ρ 0 x î ^ x + ρ 0 z î ^ z ,             ρ 1 = ρ 1 x î ^ x + ρ 1 z î ^ z
A 0 , in exp [ - j ( ρ 0 x x + ρ 0 z z ) ] .
R 0 I exp [ - j ( ρ 0 x x - ρ 0 z z ) ] ,
R 1 I exp [ - j ( ρ 1 x x - ρ 1 z z ) ] .
A 0 , out exp [ - j ( ρ 0 x x + ρ 0 z z ) ] ,
A 1 , out exp [ - j ( ρ 1 x x + ρ 1 z z ) ] .
A 0 ( z ) exp ( - j ρ 0 · r ) ,             A 1 ( z ) exp ( - j ρ 1 · r ) ,
1 + R 0 I = A 0 ( 0 ) ,
R 1 I = A 1 ( 0 ) .
H x = 1 j ω μ E y z ,
H x I = 1 ω μ { - ρ 0 z exp ( - j ρ 0 · r ) + ρ 0 z R 0 I × exp [ - j ( ρ 0 x x - ρ 0 z z ) ] + ρ 1 z R 1 I × exp [ - j ( ρ 1 x x - ρ 1 z z ) ] }
H x I I = 1 j ω μ [ exp ( - j ρ 0 · r ) ( d A 0 d z - j ρ 0 z A 0 ) + exp ( - j ρ 1 · r ) ( d A 1 d z - j ρ 1 z A 1 ) ] .
- j ρ 0 z ( 1 - R 0 I ) = d A 0 d z | z = 0 - j ρ 0 z A 0 ( 0 ) ,
+ j ρ 1 z R 1 I = d A 1 d z | z = 0 - j ρ 1 z A 1 ( 0 ) .
cos θ 0 d A 0 d z + j ɛ m β 4 A 1 = 0 ,
cos θ 1 d A 1 d z + j ɛ m β 4 A 0 = 0 ,
ρ 0 z ( 1 - R 0 I ) = ɛ m β 4 cos θ 0 A 1 ( 0 ) + ρ 0 z A 0 ( 0 ) ,
ρ 1 z R 1 I = - ɛ m β 4 cos θ 1 A 0 ( 0 ) - ρ 1 z A 1 ( 0 ) .
A 0 ( 0 ) = A 0 , in = 1 ,
R 0 I = 0 ,
R 1 I = A 1 ( 0 ) = D 10 A 0 , in = D 10 ,
D 10 = - ɛ m 8 cos 2 θ 1 .
A 1 ( 0 ) = A 1 , in ,
R 1 I = 0 ,
R 0 I = A 0 ( 0 ) = D 01 A 1 , in ,
D 01 = - ɛ m 8 cos 2 θ 0 .
[ A 0 ( 0 ) A 1 ( 0 ) R 0 I R 1 I ] = [ 1 D 01 0 - D 01 D 10 1 - D 10 0 0 D 01 1 - D 01 D 10 0 - D 10 1 ] [ A 0 , in A 1 , in R 0 I I ( 0 ) R 1 I I ( 0 ) ] ,
R 0 I I ( z ) exp [ - j ( ρ 0 x x - ρ 0 z z ) ] , R 1 I I ( z ) exp [ - j ( ρ 1 x x - ρ 1 z z ) ]
[ A 0 , out A 1 , out R 0 I I ( d ) R 1 I I ( d ) ] = [ 1 - D 01 - D 10 1 0 - D 01 - D 10 0 ] [ A 0 ( d ) A 1 ( d ) ] ,
A 0 ( d ) = A 0 , in cos ν ,             A 1 ( d ) = - j A 0 , in ( cos θ 0 cos θ 1 ) 1 / 2 sin ν ,
ν = β ɛ m d 4 ( cos θ 1 cos θ 0 ) 1 / 2 .
a + b = 2 d cos θ 1 .
R 1 I = - ɛ m 8 cos 2 θ 1 [ 1 - cos ( ν ) exp ( - j 2 β d cos θ 1 ) ] ,
R 0 I = - j ɛ m 8 cos 2 θ 0 sin ( ν ) exp ( - j 2 β d cos θ 0 ) .

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